In a CRR model, find the Initial investment of the hedging strategy

Question

Given a Cox-Ross-Rubinstein model with $T=10$, $u=1.1$, $d=0.9$, $r=0.02$, $S_0=100$ and a European call option with Strike $K=220$, find the initial investment of the hedging strategy.

I know how to compute the fair price at $t=0$ for the European call, say it is $x$, and I know that for the initial investment $(\alpha_0,\beta_0)$ of the Hedging strategy where $\alpha_0$ is the weight on the stock and $\beta_0$ on the bond, we have $x=100\alpha_0+\beta_0$. But there needs to be another equation such that I can solve it.

Answer 1

I personally think the phrase "initial investment" is a bit ambiguous, and I would have assumed it meant "[currency] value of the hedging portfolio", which you correctly note is equal to the fair value of the option.

Another possibility, which it seems like is the one you're looking for, is finding $(\alpha_0, \beta_0)$ themselves. In finding the price of the option at time 0, you are effectively finding these also. In particular, you can explicitly solve: $$100u\alpha_0 + (1 + r) \beta_0 = \text{Value of the option at t=1 if the stock goes up}$$ $$100d\alpha_0 + (1 + r) \beta_0 = \text{Value of the option at t=1 if the stock goes down}$$

And note that there should be no inconsistency with the equation you have provided, $100 \alpha_0 + \beta_0 = x$.